In this paper, we aim to study some algebraic properties of generalized bicomplex number. We also give the definitions of the generalized bicomplex Polynomial, generalized bicomplex exponential function and Generalized bicomplex cosine and sine function in order to establish our subsequent study.
The aim of this paper is the characterization of biharmonic curves in the $3-$ dimensional Lorentz-Heisenberg group $\mathcal{H}_{3}$ endowed with a left invariant Lorentzian metric $g_{i}$ in the two following cases: \begin{enumerate} \item $g_{1}=-dx^{2}+dy^{2}+(xdy+dz)^{2},$ \item $g_{2}=dx^{2}+dy^{2}-(xdy+dz)^{2}.$ \end{enumerate}
This work is concerned with the construction of minimal and maximal solutions for a class of quasilinear fourth order impulsive differential equation with integral boundary conditions, where the nonlinearity is a continuous function depending on the second and third derivative of the unknown function. We also give an example to illustrate our results.
The Klein 4-group, denoted by $V_4$ is an abelian group of order 4. It has elements $V_4 =\{0, a, b, c\},$with $a+a=b+b=c+c=0$ and $a+b=c,b+c=a,c+a=b.$ A graph $G = (V (G),E(G)),$ withvertex set $V(G)$ and edge set $E(G),$ is said to be $star$ magic if there exists a labeling $f:V(G)\rightarrow V_4\backslash\{0\}$ such that the induced mapping $V_f^+:V(G)\rightarrow V_4$ defined by $V_f^+(v)=\sum_{u\in N(v)} f^*(uv),$ where $f^*(uv)=f(u)+f(v)$ is a constant map. If this constant is $p,$ where $p$ is any non zero element in $V_4$, then we say that $f$ is a $p-star$ magic labeling of $G$ and $G$ is said to be a $p-star$ magic graph. If this constant is $0,$ then we say that $f$ is a $0-star$ magic labeling of $G$ and $G$ is said to be a $0-star$ magic graph.
In this article, we use the classical fixed point theorems to prove the main results for the integral boundary value problem for Hilfer-Katugampola fractional differential equations. An Example is presented to clarify the main results at the end .
